14.29.7 problem 7

Internal problem ID [2805]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 7
Date solved : Monday, January 27, 2025 at 06:14:02 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 y+z \left (t \right )\\ y^{\prime }&=-x \left (t \right )-3 y-z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y-z \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 72

dsolve([diff(x(t),t)=0*x(t)+2*y(t)+1*z(t),diff(y(t),t)=-1*x(t)-3*y(t)-1*z(t),diff(z(t),t)=1*x(t)+1*y(t)-1*z(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-t} t +c_1 \,{\mathrm e}^{-2 t} \\ y &= -c_2 \,{\mathrm e}^{-t}-c_3 \,{\mathrm e}^{-t} t +c_3 \,{\mathrm e}^{-t}-c_1 \,{\mathrm e}^{-2 t} \\ z \left (t \right ) &= {\mathrm e}^{-t} \left (c_3 t +c_2 -c_3 \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 87

DSolve[{D[x[t],t]==0*x[t]+2*y[t]+1*z[t],D[y[t],t]==-1*x[t]-3*y[t]-1*z[t],D[z[t],t]==1*x[t]+1*y[t]-1*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-2 t} \left (e^t (c_1 (t+1)+c_2 t+c_2+c_3)-c_2-c_3\right ) \\ y(t)\to e^{-2 t} \left (-e^t ((c_1+c_2) t+c_3)+c_2+c_3\right ) \\ z(t)\to e^{-t} ((c_1+c_2) t+c_3) \\ \end{align*}